The short answer, before I give you all the qualifiers needed to make sense of it, is 2035, give or take 7 years.
The curve above will take some explaining. But first some other important clarifications:
Often, people don't distinguish between types of ice. So you'll hear them talk about 'ice is growing', when what they mean is the center of the Greenland ice cap, or Antarctic sea ice. My comment is specific to Arctic sea ice.
Another trap people fall in to is not paying attention to what sort of statement about ice is being made. There are two parts to this. I'm referring to sea ice extent, not area (well, at 0 extent you also have 0 area, but it's still something to keep in mind). Also, I'm referring to the monthly average for September. If some day showed zero ice cover before my 2035, give or take, that doesn't disprove the prediction. It takes a solid calendar month, September, of no ice to support or refute the prediction.
Then there's the fact that it's a probabilistic prediction. 2035 is the mid-point. By my estimation method, there's about a 50% chance (54%) that 2035 or some year before that will show zero ice extent for September. It's only 6% that we'd see zero ice in 2029 (or before) . And rises to 96% that we'll see zero ice (for the month) in 2042 or before. The 'or before' is important.
How I got to those predictions turns on the probability thing I mentioned in this year's sea ice estimation note, of it sometimes being easier to work with the probability of something not happening, than trying to figure out directly the chances of it happening.
The sea ice estimation note shows a best fit curve through the data and projects it in to the future. The fit has some fuzziness to it, meaning that it doesn't pass perfectly through the data points. We don't really expect it to either -- there is weather variability. The size of this mis-fit, or, conversely, weather variability, is about 0.45 million km^2.
If we pretend that the variability follows the Normal distribution, then we can use that and our predictions to estimate the probability that we'll see 0 ice in any given year. That doesn't turn out to be useful. For instance, suppose there was a 1% chance of seeing no ice this year, and 4% of seeing no ice next year. What's your chance of seeing no ice in either year? I can compute it, but it's more tedious. It get rapidly more tedious with each year you want to consider. At 30-60 years ... ouch!
But we can consider quite easily the chance of there not being ice this year and there not being ice next year, and the year after ... out however far you want to go. What makes this easy is the and in the statement. If you're dealing with combining probabilities, and they combine by 'and', to find the chance of every single one happening, you just multiply the individual probabilities together. For the example from the previous paragraph, 1% chance of seeing no ice this year means 99% chance of seeing ice this year. 3% chance of no ice, means 97% chance of ice. The probability, then of seeing ice in all (2) years is 0.99 * 0.97, for 0.96 (96%). We can then turn around and see that the chance of some year not having ice is 100% - 96%, for 4%.
You might be noticing that 4% = 1% + 3%, and thinking you can just add the percentages. Consider tossing two coins. There's a 50% chance of getting heads on the first one, and 50% chance of getting heads on the second. (conversely, 50% chance of not getting heads). If you added, you would conclude that there's a 100% chance of not getting 2 heads. (50% chance of not getting a heads on the first coin and 50% chance of not getting heads on the second). The correct thing is to multiply (0.5*0.5), which says there's a 25% chance of getting no heads. Try tossing some pairs of coins and see which method is more correct.
In the top figure, I performed this kind of calculation for sea ice for every year out to 2070. As you see, the probability of not having any year with 0 ice for all of September is very high out to 2025. After that, we start accumulating some chances.
Now I can't say that I entirely believe this predictor. It is based only on my best fit logistic curve, rather than the ensemble. And this method knows nothing about sea ice thickness, only extent. But it makes a start on helping me (at least) think about what kinds of things are plausible. Given this, I'd need some good evidence behind a prediction of no ice for the month of September in, say, 2013. That seems extraordinarily unlikely. Conversely, 2060 looks extremely late (about a million to 1 against).